I am struggling on solving the problem related to Galois theory.
Let $K/F$ be a cyclic Galois extension whose Galois group is generated by $\sigma$ of order $n$. Suppose that $\alpha \in K$ satisfies $N_{K/F} (\alpha)=1$. Prove that there exists $\beta \in K$ such that $\alpha = \beta / \sigma(\beta)$.
I tried to use the linear independence of characters, that is, the maps $1, \sigma, \ldots, \sigma^{n-1}$ are linearly independent. There must exist $\gamma \in K$ such that $\delta := \sum_{i=1}^n \sigma^i(\gamma) \neq 0$ and in this case, $\delta / \sigma(\delta) = 1$. However, I think this observation does not give me any better insight. Any ideas or comments are welcome.